461191 Discrete Math Lecture 9: Relations

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461191 Discrete MathLecture 9 Relations

• San Ratanasanya
• CS, KMUTNB

Adpated from Dr. Goanchanart, RSU
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Todays topics

• Review of Recurrence Relations
• Administrivia
• Relations
• Properties of Relations
• n-ary Relations
• Representing Relations
• Closures of Relations
• Equivalence Relations
• Partial Orderings

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Recurrence Relations

• Recurrence relation ?????? ????? an
  ??????????????? an ???????????????????????????????
  ?????????? ?????? n n0 ??? n0 ???? Nonnegative
  Integer ???????????????????????????????????
  (Solution) ??? relation ??????????
  ???????????????????????????????????????????
• Relation ???? unique solution ????????????????????
  initial condition
• ???????????????????????????????????????????
  Recurrence Relation ???

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Examples

• ??????????????????????????????? 1 ??? ??? 5 ???
  ???? recurrence relation ?????????????????????????
  ?? ??????????????????????????????????
• ??? Pn ?????????????????????? n ???
• P1 1, P2 1, P3 1, P4 1, P5 2
• P6 3, P7 4, P8 5, .
• ?????????? P6 3 2 1, P7 4 3 1, P8 5
  4 1,
• Pn Pn-1 Pn-5

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Solving Linear Recurrence Relations

• ????????????????????? Recurrence relation
  ?????????????????????????? recursive ???
  ??????????????????????????????????????????????????
  ????????? n
• ?????????????????????????? ?????????????????????
  ????????????????????????????????????????? (Linear
  equation)
• ??????????????????????????????? 2 ?????? ???
• Homogeneous
• Non-Homogeneous

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Examples

• ????????????????????????????????? an an-1
  2an-2, a0 2, a1 7
• ??? r2 r 2 0, r1 2, r2 -1
• ??????? an a12n a2(-1)n
• ????????? initial condition ?????????????????????
  a1 3 ??? a2 -1
• ??????????? an 32n -(-1)n

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Generating Functions

• ????????????????? (Generating Function)
  ??????????????????????????? Power Series
• ???????????
• ?????????? Explicit Formula ??? Recurrence
  Relation
• ???

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Examples

• ??????????????? recurrence relation ak 3ak-1
  ?????? k 1,2,3 ????? initial condition a0 2
• ??? G(x) ???? generating function ??? ak
  ???????
• ?????????
• ?? G(x) ??? xG(x) ????????? recurrence relation
  ????????
• ???? G(x) 3xG(x) (1-3x)G(x) 2 ??????? G(x)
  2/(1-3x) ???
• ?????
• ???????

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Divide-and-Conquer and Recurrence Relations

• ???????? Recursive Algorithm ???????????????? n
  ????????????????? ????? a ????????????????????????
  ??????? n/b ?????? n ???b ????????????????????????
  ????????????????????????????? g(n)
  ??????????????????????
• ????????????????????????????????? Divide
  (???????) ?????????????????????????????
  ??????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ????? ????????????????????????????????????????????
  ??????????????????????????????????????????????????
  Conquer (????????)
• ???????? f(n) ????????????????????????????????????
  ???????? n ???????? f(n) ???????????????
  Recurrence Relation ??????
• f(n) af(n/b) g(n)
• ??????????????????? Divide-and-Conquer Recurrence
  Relation
• ???????????? Master Theorem ??????????????????????
  ?????? algorithm ????????????

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Example

• ??? f(n) f(n/3) 1 ????? n ??? 3 ????? ??? f(1)
  1 ???? f(27)
• f(27) f(9) 1 f(3) 2 f(1) 3 4

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Inclusion-Exclusion
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Example

• ????????????????????????????????? 200 ????
  Principle of Inclusion-Exclusion
• 40

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Administrivia

• Midterm Exam
• Statistic Summary
• n S1S2-MA 20421-2481-3 93
• Max ??
• Mean ??
• Min 6
• Standard Deviation ??
• Homework 1-6 and Programming Assignment 1 due
  today
• Programming Assignment 2 dues next week

Sorry, I cannot finish grading it by today
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Doubts in Midterm Exam

• 10. ??????????????????????????????????????????????
  1, 2, 3 ???????? 1 ???????? 3
• ???????? 1 ???????? 3 ??????????? 1 ??? 3
  ?????????? ????? 1 ?????????????
• ????? 1 ???????? 3 ????????????????? 1
  ?????????????? 3 ????? 1 ????????????????????
• 16. ????????????????????????????
  ??????????????????????????????? 50 ???
  ????????????????????? 10 ???? ??????????? 85 ????
  ??????????????????????????????????????????????????
  ???????????????????????? ?????????????????????????
  ?????????????? 2 ??????????????????????
• ???????? ????????????????????? 2
  ????????????????????... ?????????????????????????
  ???????? 2?
• ????? ?????????? pigeonhole ?N/k? 2 ????? k
  ??????????????? N ???????????????????????????
  ???????? N k 1

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Doubts in Midterm Exam

• 18. ??????????.???????? 3 ??? ?????? 2 ??? ???
  ????? 2 ??? ?????.????????????? 1 ???? ??????
  1??? ????????????????????????????????????
• ???????? ?????????????????????????
  ???????????????????????????????????
• ????? ???????????????????????? r ?????????????
  ?????????????????????????????????
  ????????????????????????????????????????????
• 19. ???????????????? CS ?????????????????????
  ????????????????????????????????????????? 40
  ??????????????????????????????? 10 ??
  ??????????????????????????????????????????????????
  ?????
• ???????? ????????????????????????????????????
  ?????????????????????????
• ????? ???????????????????????????????????????????
  ?????????? ?????????????????????????????????
  ????????????????????????

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Relations

• ????????
• A ??????????????????
• B ?????????????????
• R ????????????????????? (a,b) ???????????
  ???????? a ?????????????????? b
• ????? ?????????????????? CS01
• (?????, CS01)
• ??????? ?????????????????? CS02
• (???????, CS02)
• ???????? ?????????????????? CS01
• (????????, CS01)

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Relations

• ????????
• A ????????????????????????????
• B ??????????????????
• R ?????????????????????????????????????? (a,b)
  ????? a ????????? b
• ???????????? (a,b) ????????? R

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Relations

• Relations (????????????) ????????????????
  ??????????????????????????? set
  ??????????????????????????????????????????????????
  ????????? ???? ??????????????????
  ??????????????????????????????????? ???????
• Binary Relation ??????????????????????? Set 2 set
• ????? Relation ?????????????????? Binary Relation
• n-ary Relation ??????????????????? set ??????????
  2 set
• Ordered Pair ?????????????????????????????????????
  ?? Set 2 set ?????????????????? ????????????
  subset ???????? Cartesian ??????? 2 set ???????
• ??????????????? a R b ??? relation R ??? a ?? b
  ??? a R b ??????? a ??? b ????? relation R ??????
• ??????? a R b ??????? (a,b) ?R ???? a ????????
  (related) ??? b ??? R

Definition 1 ??? A ??? B ???? Set
???????? Binary Relation ??? A ?? B ???? subset
??? A x B
Definition 2 Relation ?? Set A ??? Relation ???
A ?? A ??????????? Subset ??? A x A
/
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Examples

• A 0, 1
• B a, b
• A x B
• (0,a), (0,b), (1,a), (1,b)
• R1 (0,a), (0,b)
• R2 (0,a), (1,a)
• R3 (0,b), (1,a), (1,b)
• R4 (0,a), (0,b), (1,a), (1,b)
• ?????????????????????? relation ??? A ????? B

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Examples

• ??? A 0, 1, 2 ??? B a, b ???? (0,a),
  (0,b), (1,a), (2,b) ???? relation ??? A ?? B
• ??? A 1, 2, 3, 4 ???? Ordered pair ?????????
  
  Relation R (a, b) a divides b
• R (1, 1), (1, 2), (1, 3), (1, 4), (2, 2),
  
  (2,4), (3,
  3), (4, 4)

???????????????????? ?????????????????????????
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Example

• ?????????????????????????
• R1 (a,b) a b
• R2 (a,b) a gt b
• R3 (a,b) a b or a -b
• R4 (a,b) a b
• R5 (a,b) a b 1
• R6 (a,b) a b 3

• ????????????????????????????????????
• (1,1)
• (1,2)
• (2,1)
• (1, -1)
• (2,2)

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Properties of Relations

• R ????????????????? (Reflexive) ??? a R a
  ?????????? a ?? A
• R ????????????????? (Symmetric) ??? a R b
  ????????? b R a
• R ???????????????????? (Anti-symmetric) ??? a R b
  ??? b R a ????????? a b
• R ?????????????????? (Transitive) ??? a R b ??? b
  R c ????????? a R c
• ?? Definition 3-5 ?? section 8.1

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Examples

• Relation ?? 1,2,3,4 ?????????? ???? reflexive,
  symmetric, antisymmetric, ??? transitive ????????
  ?
• R1 (1,1), (1,2), (2,1), (2,2), (3,4), (4,1),
  (4,4)
• R2 (1,1), (1,2), (2,1)
• R3 (1,1), (1,2), (1,4), (2,1), (2,2), (3,3),
  (4,1), (4,4)
• R4 (2,1), (3,1), (3,2), (4,1), (4,2), (4,3)
• R5 (1,1), (1,2), (1,3), (1,4), (2,2), (2,3),
  (2,4), (3,3), (3,4), (4,4)
• R6 (3,4)
• Reflexive R3, R5
• Symmetric R2, R3
• Antisymmetric R4, R5, R6
• Transitive R4, R5, R6

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More on Relations

• ????????????????????????? Set ????????????????????
  ?? Set Operations ??? Relations ???
• Functions ??????????????????????????? ???????
  Functions ??????? subset ??? Relations
• Composite Relation ???? ??????????????????
  ???????????? S?R ?????????????????????????????????
  ???? a R b ??? b S c ??????? a S?R c (??
  Definition 6, Sec. 8.1)
• Power ??? Relation (Rn) ??????????????????????????
  ??????????????????? R ????? n ?????
  ????????????????????????????? Recursive ?????????
  (?? Definition 7, Sec. 8.1)
• R1 R, Rn1 Rn?R, n 1, 2, 3,

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????????????????????????????????
Relation
Function
?
?
One-to-one
?
?
One-to-many
?
?
Many-to-one
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Examples

• ??? A 1,2,3 ??? B 1,2,3,4 ????? Relation
  R1 (1,1), (2,2), (3,3) ??? R2 (1,1),
  (1,2), (1,3), (1,4) ????
• R1 ? R2
• R1 ? R2
• R1 - R2
• R2 R1
• ???? Composite ??? R ??? S ?????? R ???? relation
  ??? 1,2,3 ????? 1,2,3,4 ?????? R (1,1),
  (1,4), (2,3), (3,1), (3,4) ??? S ???? relation
  ??? 1,2,3,4 ????? 0,1,2 ?????? S (1,0),
  (2,0), (3,1), (3,2), (4,1)
• S?R (1,0), (1,1), (2,1), (2,2), (3,0), (3,1)

(1,1), (1,2), (1,3), (1,4), (2,2), (3,3)
(1,1)
(2,2), (3,3)
(1,2), (1,3), (1,4)
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Example

• ??? R (1,1), (2,1), (3,2), (4,3) ???? Rn
• R2 R?R (1,1), (2,1), (3,1), (4,2)
• R3 R2?R (1,1), (2,1), (3,1), (4,1)
• R4 R3?R (1,1), (2,1), (3,1), (4,1)
• Rn R3, n 3, 4, 5,

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???????? Relations ?? Set ??????????? n ???

• ??? set A ???????? n ???.
• ?????????????? set A ??? subset ??? A x A.
• ???? A x A ????????????? n2 ???.
• Set ??????????? m ??????? 2m subset.
• ??????? A x A ???? subset ??????? 2n2.
• ???????????? 2n2 relations ?? set ??????????? n
  ???.
• Example
• set a, b, c ???????? 232 29 512
  relations.

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Example

• ???? Reflexive relation ??????????? set
  ??????????? n ???
• Reflexive ??? (a, a) ?R ?????????? Product rule
  ???? n(n-1) ordered pair
• ??????????????? Relations ?????????? 2n(n-1) ???

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n-ary Relations and its applications

• ??????????????????????? set ?????????? 2 set
  ?????? ?????????????????????
• ???? ????????. ??????????????????????? ??. ??????
  ???????????????????, ???????????????
  ???????????????????? ???????? ??????
  ?????????????????????????????? ???????
• ??? A1, A2, , An ???? set ???? n-ary Relation ??
  set ?????????????? subset ??? A1?A2??An
  ?????? A1, A2, , An ???? Domain ??? Relation ???
  n ?????? degree (?? Definition 1, Sec. 8.2)
• Example ??? R ???? relation ?? N?N?N ??????????
  triple (a, b, c) ?????? a lt b lt c ??????? (1, 2,
  3)?R ??? (2, 4, 3)?R ?????? Degree ?????? 3 ???
  Domain ??? N

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Database and Relations

• ??????????????????????????????????????????????????
  ??????????
• ?????????????????????????????????????????????????
• ??????????????????????????????????? ????????????
  ?? ????? ?????? ??????????????????????????????????
  ???????????????????????
• ?????????????????????????????????????????????????
• ??????????????????????????????????????????????????
  ??????????????????????????? ???? Relational Data
  Model (RDM) ??????????????? IBM ?????????????????
• ??????????????????????????????????????????????????
  ????? Relational DataBase Management System ????
  RDBMS
• ???????????????????????????????????????????????
  SQL ???? SEQUEL ???????????? Structured English
  Query Language
• ????????????????????????????????????? ????
  Oracle, MS Access, mySQL ???????

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Relational Database

• ?????????? record (???????) ???????? n-tuples
  ??????????????? fields ???? ???????????? ????
  ??????
• ???? ?????????. ???? record ??????????????????????
  ?????????. ???? ???? ??????? ??????. ???????? GPA
  ???????
• ?? RDM ??????????? record ??????????????????????
  n-ary relation
• ?????????????????????????? ????? (Table)
  ????????????????????????????????????
  ??????????????????????????? Table

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Relational Database
??????????????????????????????????????????????

• Primary key ??? Domain (???? Field)
  ???????????????? n-tuple ?? Table ???
• ???????????? Domain ????????????? Primary key ???
  ????????????????????????? n-ary relation
  ??????????????????????????????????????? ??
  ???????????????????????? ???????? Table
• Composite key ??? Cartesian product ??? Domain
  ???????????? ?????????? n-tuple ??????????
• ???? Domain ??? Student_Name ??? Major

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Operations on n-ary Relations

• ????????????????????????????????? RDB
  ??????????????????????? ??????????????????????????
  ?????????????????????????????
• ????????????????? ?????????????????????????
  Condition ??? ???????? ???????????????? n-tuple
  ??????????????? Relation ???? ???? Selection
  Operator ??????? map n-ary ?? Relation ?????
  n-tuple ????????????? Condition ???????
• ?????????????????????????????????????? ???
  Projection ?????? map ??? n-tuple ?????? m-tuple
• ???????????? Table ?????????????????? Join
  ???????? Table ???????????????????????????????????
  ????
• ?? Definiton 2-4 ?? Sec 8.2

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Operations on n-ary Relations
Definition 2 ??? R ???? n-ary Relation ??? C
???? Condition ??? Element ?? R ??????????
??????? selection operator (Sc) ??????? Map ???
n-ary Relation ?? R ????? n-ary Relation
????????????????? n-Tuple ??? R ????????????
Condition C
Definition 3 Projection Pi1i2,,im ?? Map ???
n-tuple (a1, a2, , an)????? m-tuple (a1, a2, ,
am) ?????? m ? n
Definition 4 ??? R ???? Relation ????? Degree m
??? S ???? Relation ????? Degree n ??? Join
Jp(R, S) ?????? p ? m ??? p ? n ???? Relation
????? Degree m n - p ??????????????????? (m n
p)-tuple (a1, a2, , am-p, c1, c2, , cp, b1,
b2, , bn-p) ?????? m-tuple (a1, a2, , am-p, c1,
c2, , cp) ????? R ??? n-tuple (c1, c2, , cp,
b1, b2, , bn-p) ????? S
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Examples

• ????????? C1 (Major Computer Science ? GPA
  gt 3.5) ???????????????? n-ary Relation ?? Table 1
  ????????????????????
• ??? 4-tuple ??????????? (Ackermann, 231455,
  Computer Science, 3.88)
• ?????? Projection P1,3 ??? 4-tuples (2,3,0,4),
  (Jane Doe, 234111001, Geogrpahy, 3.14), (a1, a2,
  a3, a4) ????????????????????????
• ??? (2,0), (23411101, Geography), (a1, a3)
• ??????????????????? Projection P1,4 ??? Table 1
  ???????
• ??????????? Join Table 5 ??? 6 ???????????????????
  ????

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Example
38
SQL Examples
SELECT Departure_Time FROM Flights WHERE
Destination Detroit

• Select ?? SQL ??????? Projection
• ????????????? Database ??? SQL ????????????? ??.
  ????? ??????????? Database

SELECT Professor, Time FROM Teaching_assignments,
Class_schedule WHERE Department Mathematics
(Rosen, 300 P.M.)
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??????????????????? (Representing Relations)

• ?????????????????????????????????????? ????
  Ordered pair ???????????????????????????????
• ??????????? ????????????????????????????????????
  Zero-One Martix ??? Directed Graph
  ?????????????????????
• ??????????? Zero-One Matrix ??????????????????????
  ?????????????????????????
• Directed Graph ???????????????????????????????????
  ?????????????????????????????????????

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Representing Relations Using Matrices

• Relation ??????? Finite Set ?????????????????
  Zero-One Matrix ????????? R ???? Relation ??? A
  ????? B ?????? Element ?? A ???B
  ?????????????????????????? ???????????????????????
  ??? ??????? Relation R ????????????????? Matrix
  MR mij ??????
• ??????? ??? Element ?? Matrix ??? Row ???
  Column ij ??????????? ????? ??? ai
  ????????????????? bj ?????????? R
  ?????????????????????????? ?????
• Example ????? A 1,2,3, B 1,2 ??? R ????
  relation ??? A ?? B ?????? R (a,b) a?A ? b?B
  ? a gt b ???? Matrix ??????? R ???????? ?????? a1
  1, a2 2, a3 3, b1 1, b2 2
• ???????? R (2,1), (3,1), (3,2) ???????

40
41
Representing Relations Using Matrices

• Matrix ??????? Relation ?? Set ???????????????????
  ???????? Relation ?????????
• Relation R ?? A ?????? Reflexive ?????????? mii
  1 ???????? ?????????????????????????? ??? Matrix
  ??????? R ???? ????????? 1 ??????
• Relation R ?? A ?????? Symmetric ?????????? mij
  mji ???????? ????????? (i,j) ??? (j,i) ??? Matrix
  ??????? R ???? ?????????????????? ???? MR MRT
• Relation R ?? A ?????? Anti-symmetric ??? mij 1
  ???? mij 0 ????? i ? j ???? ??? mij 0 ????
  mij 0 ????? i ? j

42
Representing Relations Using Matrices

• Boolean Operation ??? Matrix ???????????????
  Union ??? Intersect ??? Relation ???
• ?????????? ??????????? Composite of Relation
  ?????? Matrix ?????????? ????????? Boolean Product

43
Examples

• ??? Relation R ?? Set ???????? Matrix ????????
  ??????? R???? Reflexive, Symmetric ???/????
  Antisymmetric
• ????????? Relation R1 ??? R2 ?? Set A ????????
  Matrix ???????? ???? Matrix ??????? R1 ? R2 ???
  R1 ? R2

R ?????? Reflexive ??? Symmetric ??????????
Antisymmetric
44
Examples

• ???? Matrix ??????? Relation SoR ????? Matrix
  ???? R??? S ???
• ???? Matrix ??????? R2 ???????????

45
Representing Relations Using Directed Graphs

• ????????????????????? Relation ???????????????
  ??????????? Graph ??????????? Element ??? Set
  ?????????????? ???????? Ordered Pair
  ?????????????????????????????????
  ?????????????????? ????????????????????????????
  Directed Graph ???? Digraph

Definition 1 Directed Graph ???? Digraph
???????????? Set V ??? Vertices ???? Nodes
????????? Set E ??? Ordered Pair ??? Element ??
V ??????????? Edges ???? Arcs ?????? Vertex a
?????????? Initial Vertex ??? Edge (a, b) ???
Vertex b ?????????? Terminal Vertex ??? Edge ???
Edge ???????? (a, a) ???????????????????????????
????? Vertex a ??????????????? ???? Edge
??????????? ??????????? Loop
45
46
Representing Relations Using Digraphs

• ???????????? Directed Graph ??????????????????????
  ?????????? Relation ??? ????
• ??? Relation ???? Reflexive ???? ???????? Loop
  ?????? Vertex ??? Directed Graph ????
• ??? Relation ???? Symmetric ???? ??????? Edge
  ???????????????? Vertex ???????????????
  ??????????? Edge ??????????????????????????
• ??? Relation ???? Antisymmetric ??????????? Edge
  ???(?????????) ???????? Vertex ??????????
• ??? Relation ???? Transitive ??????????? ?????
  Edge ??? Vertex x ????? y ?????? Vertex y ????? z
  ???? ???????? Edge ??? Vertex x ????? z ????

46
47
Examples

• ????? Directed Graph ????????????? Vertex a, b,
  c, ??? d ??? Edge (a, b), (a, d), (b, b), (b,
  d), (c, a), (c, b), ??? (d, b)
• ???? Ordered Pair ?? Relation R ???????????
  Directed Graph ????????

48
Example

• ??????? Relation ??????????? Directed Graph
  ?????????????????? Reflexive, Symmetric,
  Antisymmetric ???/???? Transitive

Symmetric
49
Closures of Relations

• ????????????????? Network, ???????????????????????
  ??????????????? (Node) ???????????
  ??????????????????????????????????????? Data link
  ???????????????????? ??????????????
  ??????????????????????????????????????????????????
  ????? ???????????????????? Node
  ?????????????????????????????????????????? Node
  ???????????????? ????????????????? link
  ??????????????????????????????????????????????????
  ???????????? Relation ???????????????????????????
  ????????????????????? transitive closure ????
  ??????????????????????????

49
50
Closures of Relations

• ?????? R???? Relation ?? Set A, Relation
  R???????????????????????????? P ????????????
  Reflexive, Symmetric ???? Transitive
  ?????????????? Relation ?????????????? P ????????
  ??? S???????????? R ?????? S ?????? Subset
  ??????? Relation ?????????????? P ????? R
  ?????????? ????????????????? S ??????? Closure
  ??? R ????????????? P

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Reflexive Closure

• ??????? Relation R (1,1), (1,2), (2,1), (3,2)
  ?? Set A 1, 2, 3 ????????? R ???????
  Reflexive ?????????? Reflexive Relation
  ??????????????????? R ?????????????????????????
• ????? (2,2) ??? (3,3) ???? R ??????? R ????
  Reflexive
• ???????????? Ordered pair ??? (a, a)
  ????????????????? R
• ??????????????? Relation ????????? R ??????????
  ????????????? Reflexive Relation ??????????? R
  ?????????? ???????????????? (2,2) ??? (3,3)
• ??????? ????????? Relation R ????? (2,2) ???
  (3,3) ?????? ???? Reflexive ???????
  ???????????????? Reflexive Relation ????? R
  ?????? ???????????????? Reflexive Closure ??? R
• ???????? Relation R ?? Set A ??????????????
  Reflexive Closure ??? R?????????????????? ??????
  (a,a), a ?A ????????????????? R?????? R
• ????????????????? Reflexive Closure ??? R
  ??????????? R???????? ? ??? Diagonal Relation ??
  A ??? ? (a,a) a ?A

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Symmetric Closure

• ??????? Relation R (1,1), (1,2), (2,2), (2,3),
  (3, 1), (3,2) ?? Set A 1, 2, 3 ????????? R
  ??????? Symmetric ?????????? Symmetric Relation
  ??????????????????? R ?????????????????????????
• ????? (2,1) ??? (1,3) ???? R ??????? R ????
  Symmetric
• ???????????? Ordered pair ??? (b, a) ??? (a, b) ?
  R ?????????????? R
• ??????????????? Relation ????????????????? R
  ?????????????? Symmetric Relation ???????????? R
  ?????????? ???????????????? (2,1) ??? (1,3)
• ??????? ????????? Relation R ????? (2,1) ???
  (1,3) ?????? ???? Symmetric ???????
  ???????????????? Symmetric Relation ????? R
  ?????? ???????????????? Symmetric Closure ??? R
• ???????? Relation R ?? Set A ??????????????
  Symmetric Closure ??? R????????? union R ????
  Inverse ??? R ???? R-1
• ????????????????? Symmetric Closure ??? R
  ??????????? R? R-1 ?????? R-1 (b,a) (a,b)
  ?R

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Examples

• ???? Reflexive Closure ??? Relation R (a,b)
  a lt b ?? set ??? Integer
• ???? Symmetric Closure ??? Relation R (a,b)
  a gt b ?? set ??? Positive Integer

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Transitive Closure

• ???????? Relation R (1,3), (1,4), (2,1),
  (3,2) ?? Set A 1, 2, 3, 4 ????????? R
  ??????? Transitive ?????????? Transitive Relation
  ??????????????????? R ?????????????????????????
• ????????????? (1,2), (2,3), (2,4) ??? (3,1) ????
  R ????????????????? R ???? Transitive ???
  ???????????????????????? (3.4)
• ????????????????? ordered pair ????????????
  ?????????????????????????? R ???? Transitive
• ??????????????? Transitive Closure
  ???????????????????????? Reflexive ??? Symmetric
• ?????????? Transitive Closure????
  ??????????????????????????????????? ???? Path
  ????

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Paths in Directed Graphs

• ????????? Transitive Relation ???? Digraph
  ????????????????????? path ????

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Example

• ???????????????????????? Path ??? Directed Graph
  ????????????????????

a,b,d,e ???? Path ???????????? 3
a,e,c,d,b ??????? Path ????????? (c,d) ?????? Edge
b,a,c,b,a,a,b ???? Path ???????????? 6
d,c ???? Path ???????????? 1
c,b,a, ???? Path ???????????? 2
e,b,a,b,a,b,e ???? Path ???????????? 6
????????????? ???? 2 Path ??????? Circuit ???
b,a,c,b,a,a,b ??? e,b,a,b,a,b,e
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Paths in Directed Graphs

• ??????? Path ???????? Relation ???????????
  Directed Graph ??????????? ???? Path ??? a ?? b
  ?? R ????? Sequence ??? Element a, x1, x2,,
  xn-1,b ?????? (a, x1) ? R, (x1, x2) ? R, ,
  (xn-1,b) ? R ??????????? Theorem 1
• ??????????? Path ????????????????? Transitive ???
  Relation ????????????????????????? Vertices ??
  Directed Graph ?????? Path ?????????????????

Theorem 1 ??? R ???? Relation ?? Set A ???? Path
???????????? n ?????? n ???? Positive Integer ???
a ?? b ?????????? (a,b) ? Rn
Definiton 2 ??? R???? Relation ?? Set ,
Connectivity Relation R ????????????????????
(a,b) ????? Path ????????????????????? ????? ???
a?? b ?? R
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Transitive Closure

• ????????? Rn ?????????????????? (a,b) ???????
  path ??????? n ??? a ?? b ????? R ???? union ???
  Rn ??????? ????
• ??? Theorem 2 ????????? ????? Transitive Closure
  ???????? Connectivity Relation ???????
• Example ??? R ???? Relation ?? Set
  ?????????????? ????????????????? (a,b) ??? a
  ?????? b ???? Rn ??? R ?????? n ???? positive
  integer ?????????? 1

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Transitive Closure

• ??????????????????? R ?????????????????????? R
  ??? path ?????????????????? ??????????????????????
  ?? path ?????????????????? R ????????? Lemma 1
• ??????? Lemma 1 ?????
• ??????????????? Matrix ??????? R ???
  (?????????????? Defiiniton 3 ??? Algorithm 1)
  ???????????? O(n4) (?????????????? Sec. 8.4)
• Roy Warshall ??????? algorithm ????????????? R
  ???????????????????????????? Matrix ?????????????
  O(n3)

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Warshalls Algorithm
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Example
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Washalls Algorithm (Cont.)
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Warshalls Algorithm
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Equivalence Relations
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Equivalence Relations
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Equivalence Class Partition
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Partial Orderings
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Lexicographic Ordering
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Hasse Diagram
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Minimal and Maximal
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The Greatest/The Least Element
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Upper Bound and Lower Bound
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LUB and GLB
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Lattice
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102
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Topological Sorting
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Homework 8

• Section 8.1
• 8, 9, 13, 24, 27, 33
• Section 8.2
• 10, 26, 19, 28, 29
• Section 8.3
• 31, 32
• Section 8.4
• 15, 35
• Section 8.5
• 9, 11, 15, 16
• Section 8.6
• 12, 13, 40, 64, 65
• Supplementary
• ---

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